$p$-Adic Brownian Motion is a Scaling Limit

Abstract

A $p$-adic Brownian motion is a continuous time stochastic process in a $p$-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in $(1,\infty )$, and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.

19 pages

References in corpus (6)

• p-Adic Mathematical Physics
• p-Adic Models of Ultrametric Diffusion Constrained by Hierarchical Energy Landscapes
• Application of p-adic analysis to models of spontaneous breaking of the replica symmetry
• P-adic numbers and replica symmetry breaking
• Ultrametric theory of conformational dynamics of protein molecules in a functional state and the description of experiments on the kinetics of CO binding to myoglobin
• Brownian Motion and Finite Approximations of Quantum Systems over Local Fields